Consider two independent normal distributions. A random...

Consider two independent normal distributions. A random sample of size $n_{1}=20$ from the first distribution showed $\bar{x}_{1}=12$ and a random sample of size $n_{2}=25$ from the second distribution showed $\bar{x}_{2}=14$. (a) Check Requirements If $\sigma_{1}$ and $\sigma_{2}$ are known, what distribution does $\bar{x}_{1}-\bar{x}_{2}$ follow? Explain. (b) Given $\sigma_{1}=3$ and $\sigma_{2}=4$, find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$. (c) Check Requirements Suppose $\sigma_{1}$ and $\sigma_{2}$ are both unknown, but from the random samples, you know $s_{1}=3$ and $s_{2}=4$. What distribution approximates the $\bar{x}_{1}-\bar{x}_{2}$ distribution? What are the degrees of freedom? Explain. (d) With $s_{1}=3$ and $s_{2}=4$, find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$. (e) If you have an appropriate calculator or computer software, find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$ using degrees of freedom based on Satterthwaite's approximation. (f) Interpretation Based on the confidence intervals you computed, can you be $90 \%$ confident that $\mu_{1}$ is smaller than $\mu_{2}$ ? Explain.

Consider two independent normal distributions. A random sample of size $n_{1}=20$ from the first distribution showed $\bar{x}_{1}=12$ and a random sample of size $n_{2}=25$ from the second distribution showed $\bar{x}_{2}=14$.
(a) Check Requirements If $\sigma_{1}$ and $\sigma_{2}$ are known, what distribution does $\bar{x}_{1}-\bar{x}_{2}$ follow? Explain.
(b) Given $\sigma_{1}=3$ and $\sigma_{2}=4$, find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$.
(c) Check Requirements Suppose $\sigma_{1}$ and $\sigma_{2}$ are both unknown, but from the random samples, you know $s_{1}=3$ and $s_{2}=4$. What distribution approximates the $\bar{x}_{1}-\bar{x}_{2}$ distribution? What are the degrees of freedom? Explain.
(d) With $s_{1}=3$ and $s_{2}=4$, find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$.
(e) If you have an appropriate calculator or computer software, find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$ using degrees of freedom based on Satterthwaite's approximation.
(f) Interpretation Based on the confidence intervals you computed, can you be $90 \%$ confident that $\mu_{1}$ is smaller than $\mu_{2}$ ? Explain.

Key Concepts

Histograms

Histograms are graphical representations that display the distribution of data by grouping values into bins and showing the frequency or relative frequency of observations within each bin. They provide a visual summary of the shape, spread, and central tendency of the sampling distribution, making it easier to identify patterns, trends, and anomalies in the data.

Relative Frequencies

Relative frequencies convert the raw counts from frequency tables into proportions by dividing each count by the total number of observations. This normalization facilitates comparisons across different samples and approximates the probability of each outcome, which is vital for understanding the underlying structure of sampling distributions.

Sampling Distribution

A sampling distribution represents the probability distribution of a given statistic based on a large number of samples drawn from a specific population. It provides insight into the variability of the statistic, helping us understand how it may change from sample to sample and forming the basis for inferential statistics, such as hypothesis testing and confidence intervals.

Frequency Tables

Frequency tables organize raw data by listing the number of occurrences for each distinct value or category. They serve as a foundational tool in summarizing data, allowing us to see how frequently different outcomes, including sample statistics, occur across many samples.

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